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Let $\lambda_{max}$ be a shifted maximal real eigenvalue of a random $N\timesN$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix') in the$N\to\infty$ limit. It was shown by Poplavskyi, Tribe, Zaboronski \cite{PZT}that the limiting distribution of the maximal real eigenvalue has $s\to-\infty$asymptotics $$\mathbb{P} [ \lambda_{max} < s ] = {\rme}^{\frac{1}{2\sqrt{2\pi}} \zeta(\frac32)s + \mathcal{O}(1)},$$ where $\zeta$is the Riemann zeta-function. This limiting distribution was expressed by Baik,Bothner \cite{BB18} in terms of the solution $q(x)$ of a certainZakharov-Shabat inverse scattering problem, and the asymptotics was extended tothe form $$\mathbb{P} [ \lambda_{max} < s ] = {\rme}^{\frac{1}{2\sqrt{2\pi}}\zeta(\frac32)t} c(1+_\mathcal{O}(1)),\s\to-\infty.$$ We show that $q(x)$ is a smooth function, which behaves as$\frac{1}{x}$ as $x\to-\infty.$ Second, we show that the error term in theasymptotics is subexponential, i.e. smaller that ${\rm e}^{-C|s|}$ for any $C.$Third, we identify the constant $c$ as a conserved quantity of a certain fastdecaying solution $u(x,t)$ of the Korteweg-de Vries equation. This, inprinciple, gives a way to determine $c$ via the known long-time $t\to+\infty$asymptotics of $u(x,t).$ We also conjecture a representation for the $c$ interms of an integral of the Hastings-MacLeod solution of Painlev\'e IIequation.

On the solution of the Zakharov-Shabat system, which arises in the analysis of the largest real eigenvalue in the real Ginibre ensemble